### Student Page 1.7.2 Rectangle Patterns in the 100s Chart

In Analyzing the 100s Chart you found patterns in rows, columns, or diagonals. This time your pattern search will be confined to rectangles.

#### 1.

Consider the rectangle in TableĀ 1.7.2.1. What patterns do you see in the numbers in this rectangle?

84 | 85 | 86 |

74 | 75 | 76 |

64 | 65 | 66 |

54 | 55 | 56 |

Let \(x = 84\text{.}\) In the rectangle, write the other values in the rectangle in terms of \(x\text{.}\)

##### (a)

Use your relabeling to justify any numeric patterns you found.

##### (b)

Find another pattern in the relabeled rectangle.

##### (c)

Does your pattern work for a 4 by 3 rectangle with upper left corner labeled 51?

##### (d)

Does your pattern work for any other 4 by 3 rectangles? Why or why not?

Work on the the following exercises alone for at least 10 minutes. When each member of your group has found at least one rectangle pattern, share the patterns you found with your group.

#### 2.

Using a different color to outline each rectangle, draw three rectangles following the lines in TableĀ 1.6.3.1 (reprinted below) with the following rules:

100 | 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 |

90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 |

80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 |

70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 |

60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 |

50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 |

40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 |

30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 |

20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |

10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

##### (a)

Make sure all three rectangles have different dimensions.

##### (b)

Make one rectangle large and one rectangle small.

##### (c)

Make only one square.

#### 3.

What patterns do you notice among the numbers in one of the rectangles? Find at least three patterns. Each pattern you find must not extend beyond the rectangle.

#### 4.

Of the patterns you found, do any work for all three rectangles? Explain.

#### 5.

Choose one of the patterns you found. Explain why the pattern works.

#### 6.

Challenge yourself to use variables to show why the pattern works.

As a group, choose your 3 favorite patterns, including at least one that you think other groups won't find. Share your patterns with the class. As a group, choose 2 patterns shared by other groups to justify. Use variables and algebra to justify that each pattern works.